Brush tomato slices, avocado halves and onion slices with oil; season with salt and pepper to taste. Place onions on grill; grill 8 to 10 minutes or until softened. Grill tomato slices and avocado halves 3 to 4 minutes per side.

Meanwhile, in a small bowl combine mayonnaise and basil. Season with salt and pepper to taste; set aside.

Spread a thin layer of basil-mayonnaise on each cut side of bread. Slice the avocado and place slices on bottom halves of bread. Top with slices of onion, tomato and cheese. Cover with top halves of bread, and serve.